2. Basics of Multirate Systems

2.1 Frequency-Domain Characterisation of Down-Sampling

Downsampler's response to the input sequence :

where .

Example 2.1

F = [0,0.1,0.46,1]; A=[0,1,0,0]; % Setting the input parameters for fir2

x = fir2(256,F,A); % Generating the original signal 'x'

X = fft(x,1024); % Computing the spectrum of the original signal

f = 0:1/512:(512-1)/512; % Normalized frequencies

figure

%subplot(4,1,1)

plot(f,abs(X(1:512)))

text(0.7,0.7,'original')

ylabel('|X(e^j^\omega)|'),xlabel('\omega/\pi')

title('Spectrum of the original signal')

M = 2; % Down-sampling factor

y2 = x(1:M:256); % Down-sampling

Y2 = fft(y2,1024); % Computing the spectrum of the down-sampled signal

figure

plot(f,abs(Y2(1:512)))

text(0.6,0.35,'downsampled, M=2')

ylabel('|Y_2(e^j^\omega)|'),xlabel('\omega/\pi')

title('Spectrum of down-sampled signal, M=2')

M = 3; % Down-sampling factor

y3 = x(1:M:256); % Down-sampling

Y3 = fft(y3,1024); % Computing the spectrum of the down-sampled signal

figure

plot(f,abs(Y3(1:512)))

text(0.5,0.3,'downsampled, M=3')

ylabel('|Y_3(e^j^\omega)|'),xlabel('\omega/\pi')

title('Effect of aliasing, M=3')

M = 4; % Down-sampling factor

y4=x(1:M:256); % Down-sampling

Y4 = fft(y4,1024); % Computing the spectrum of the down-sampled signal

figure

plot(f,abs(Y4(1:512)))

text(0.5,0.2,'downsampled, M=4')

ylabel('|Y_3(e^j^\omega)|'),xlabel('\omega/\pi')

title('Effect of aliasing, M=4')

disp('END OF EXAMPLE 2.1')

END OF EXAMPLE 2.1

2.2 Frequency-Domain Characterisation of Up-Sampling

Example 2.2

F = [0,0.2,0.9,1]; A = [0,1,0,0]; % Setting the input parameters for fir2

x = fir2(128,F,A); % Generating the original signal ‘x’

X = fft(x,1024); % Computing the spectrum of the original signal

f = 0:1/512:(512-1)/512; % Normalized frequencies

figure

%subplot(4,1,1),

plot(f,abs(X(1:512)))

ylabel('|X(e^j^\omega)|')

text(0.6,0.6,'original')

title('Spectrum of the original signal')

L = 2; % Up-sampling factor

y2 = zeros(1,L*length(x));

y2([1:L:length(y2)]) = x; % Up-sampled signal, L=2

Y2 = fft(y2,1024); % Computing the spectrum of the up-sampled signal

plot(f,abs(Y2(1:512)))

ylabel('|Y_2(e^j^\omega)|'),xlabel('\omega/\pi')

text(0.8,0.5,'L=2')

title('Spectrum of the upsampled signal')

L = 3; % Up-sampling factor

y3 = zeros(1,L*length(x));

y3([1:L:length(y3)]) = x; % Up-sampled signal, L=3

Y3 = fft(y3,1024); % Computing the spectrum of the up-sampled signal

plot(f,abs(Y3(1:512)))

ylabel('|Y_3(e^j^\omega)|'),xlabel('\omega/\pi')

text(0.75,0.5,'L=3')

title('Spectrum of the upsampled signal')

L = 4; % Up-sampling factor

y4 = zeros(1,L*length(x));

y4([1:L:length(y4)]) = x; % up-sampled signal, L=4

Y4 = fft(y4,1024); % Computing the spectrum of the up-sampled signal

plot(f,abs(Y4(1:512)))

ylabel('|(Y_4(e^j^\omega)|'), xlabel('\omega/\pi')

text(0.7,0.5,'L=4')

title('Spectrum of the upsampled signal')

disp('END OF EXAMPLE 2.2')

END OF EXAMPLE 2.2

2.3 Spectrum of Decimated Signal

The role of H(z) is to bandlimit the spectrum of the original signal to .

Example 2.3

X = fft(x,1024); %

f = 0:1/512:(512-1)/512; % Normalized frequencies

figure

plot(f,abs(X(1:512)))

ylabel('|X(e^j^\omega)|')

text(0.8,0.8,'original')

title('Spectrum of the original signal')

M = 2; % Down-sampling factor

y = x(1:M:256); % Down-sampling

Y = fft(y,1024); % Computing the spectrum of the down-sampled signal

figure

plot(f,abs(Y(1:512)))

ylabel('|Y(e^j^\omega)|')

text(0.65,0.5,'downsampled M=2')

title('Spectrum of the downsampled signal')

yd = decimate(x,M); % Decimated signal

Yd = fft(yd,1024); % Computing the spectrum of the decimated signal

figure

plot(f,abs(Yd(1:512)))

xlabel('\omega/\pi'),ylabel('|Y_d(e^j^\omega)|')

text(0.6,0.3,'decimated, M=2')

title('Spectrum of the interolated signal')

disp('END OF EXAMPLE 2.3')

END OF EXAMPLE 2.3

2.4 Spectrum of Interpolated Signal

The role of H(z) is to bandlimit the spectrum of the upsampled signal to , and thus eliminate unwanted spectra produced by upsampling.

Example 2.4

F = [0,0.2,0.9,1]; A = [0,1,0,0]; % Setting the input parameters for fir2

x = fir2(128,F,A); % Generating the original signal ‘x’

X = fft(x,1024); % Computing the spectrum of the original signal

f = 0:1/512:(512-1)/512; % Normalized frequencies

figure

plot(f,abs(X(1:512)))

ylabel('|X(e^j^\omega)|'),xlabel('\omega/\pi')

text(0.6,0.7,'original')

title('Spectrum of the original signal')

L = 4; % Up-sampling factor

xu = zeros(1,L*length(x));

xu([1:L:length(xu)]) = x; % Up-sampled signal

Xu = fft(xu,1024); % Computing the spectrum of the up-sampled signal

figure

plot(f,abs(Xu(1:512)))

ylabel('|X_u(e^j^\omega)|'),xlabel('\omega/\pi')

text(0.62,0.8,'upsampled L=4')

title('Spectrum of the upsampled signal')

y = interp(x,L); % Interpolated signal

Y = fft(y,1024); % Computing the spectrum of the up-sampled signal

figure

plot(f,abs(Y(1:512)))

ylabel('|(Y(e^j^\omega)|'), xlabel('\omega/\pi'),

text(0.5,2,'interpolated L=4')

title('Spectrum of the interpolated signal')

figure

subplot(3,1,1), stem(54:74,x(55:75))

ylabel('x[n]')

text(68,0.35,'original')

title('Time Domain Prezentation')

subplot(3,1,2)

stem(L*55-1:L*74-1,xu(L*55-1:L*74-1))

ylabel('x_u[m]')

text(270,0.33,'upsampled L=4')

axis([L*55-1,L*74-1,-0.5,0.5])

subplot(3,1,3)

stem(L*55-1:L*74-1,y(L*55-1:L*74-1))

ylabel('y[m]'), xlabel('Time index')

text(270,0.33,'interpolated L=4')

axis([L*55-1,L*74-1,-0.5,0.5])

disp('END OF EXAMPLE 2.4')

END OF EXAMPLE 2.4

2.5 The Six Identities

First identity

Second identity

Third Indentity

Example 2.5

clear all, close all

n = 0:60; % Time index

x = cos(2*pi*0.05*n); % Generating the original signal

h = fir1(10,0.5); % Designing the filter transfer function H(z)

hu = upsample(h,4); % Transfer function H(z^4)

y1 = filter(hu,1,x); % Filtering

y = downsample(y1,4); % Down-sampling

m = 0:length(y)-1; % Time index

figure

subplot(3,2,1), stem(n,x), ylabel('x[n]')

title('Figure (a): x[n],y_1[n],y[m]')

subplot(3,2,3), stem(n,y1), ylabel('y_1[n]')

subplot(3,2,5), stem(m,y), ylabel('y[m]')

xlabel('Time index')

y2 = downsample(x,4); % down-sampling

y = filter(h,1,y2); % filtering

subplot(3,2,2), stem(n,x),ylabel('x[n]')

title('Figure (b): x[n],y_2[m],y[m]')

subplot(3,2,4), stem(m,y2), ylabel('y_2[m]')

subplot(3,2,6), stem(m,y),ylabel('y[m]')

xlabel('Time index')

disp('The third identity verified')

The third identity verified

disp('END OF EAMPLE 2.5')

END OF EAMPLE 2.5

Fourth Identity

Fifth Identity

Sixth Identity

Example 2.6

clear all, close all

n = 0:15; % Time index

x = cos(2*pi*0.1*n); % Generating the original signal

h = fir1(10,0.5); % Designing the filter transfer function H(z)

hu = upsample(h,2); % Transfer function H(z^2)

y1 = filter(h,1,x); % Filtering

y = upsample(y1,2); % Up-sampling

m = 0:length(y)-1; % Time index

figure

subplot(3,2,1), stem(n,x), ylabel('x[n]')

title('Figure (a): x[n],y_1[n],y[m]')

subplot(3,2,3), stem(n,y1),ylabel('y_1[n]')

subplot(3,2,5), stem(m,y), ylabel ('y[m]')

xlabel('Time index')

axis([0,30,-1,1])

y2 = upsample(x,2); % Up-sampling

y = filter(hu,1,y2); % Filtering

subplot(3,2,2), stem(n,x), ylabel('x[n]')

title('Figure (b): x[n],y_2[m],y[m]')

subplot(3,2,4), stem(m,y2), ylabel('y_2[m]')

axis([0,30,-1,1])

subplot(3,2,6), stem(m,y), ylabel('y[m]')

xlabel('Time index')

axis([0,30,-1,1])

disp('The sixth identity verified')

The sixth identity verified

disp('END OF EXAMPLE 2.6')

END OF EXAMPLE 2.6

2.6 Cascading Sampling-Rate Alteration Devices

Example 2.7

clear all, close all

n = 0:15; % Time index

L = 2; M = 3; % Up-sampling and down-sampling factors

x = cos(2*pi*0.1*n); % Generating the original signal

v1 = upsample(x,L); % Up-sampling

y = downsample(v1,M); % Down-sampling

r = 0:length(v1)-1; % Time index

m = 0:length(y)-1; % Time index

figure

subplot(3,2,1), stem(n,x), ylabel('x[n]')

title('Figure (a): x[n],v_1[r],y[m]')

subplot(3,2,3), stem(r,v1), ylabel('v_1[r]')

axis([0,30,-1,1])

subplot(3,2,5), stem(m,y), ylabel('y[m]')

xlabel('Time index')

v2 = downsample(x,M); % Down-sampling

r = 0:length(v2)-1; % Time index

y = upsample(v2,L); % Up-sampling

m = 0:length(y)-1; % Time index

subplot(3,2,2), stem(n,x), ylabel('x[n]')

title('Figure (b): x[n],v_2[r],y[m]')

subplot(3,2,4), stem(r,v2), ylabel('v_2[r]')

axis([0,5,-1,1])

subplot(3,2,6), stem(m,y), ylabel('y[m]')

xlabel('Time index')

axis([0,10,-1,1])

disp(['Equivalence satisfied since L and M are relativly prime.'])

Equivalence satisfied since L and M are relativly prime.

disp('END OF EXAMPLE 2.7')

END OF EXAMPLE 2.7

2.7 Sampling-Rate Conversion with the Phase Offset

where for k = 0, the resulting signal is the down-sampled signal without the phase offset. Each value of k defines a new signal.

In the next example, we demonstrate the effects of the phase offset to the spectrum of the decimated signal. We show that when the spectrum of the original signal is bandlimitted to , there is no aliasing in the signals produced by M-fold decimation. However, when the spectrum of the original signal exceeds this limit, down-sampling opperation produces the aliasing.

Example 2.8

M = 2; % Down-sampling factor

y0 = downsample(x,M); % down-sampling without phase offset

Y0 = fft(y0,1024); % Computing the signal spectrum

subplot(3,2,3), plot(f,abs(Y0(1:512))), text(0.8,0.7,'(b)')

ylabel('|Y_0(e^{j\omega})|'), axis([0,1,0,1])

y1 = downsample(x,M,1); % Down-sampling with the phase offset

Y1 = fft(y1,1024); % Computing the signal spectrum

subplot(3,2,5), plot(f,abs(Y1(1:512))), text(0.8,0.7,'(c)')

ylabel('|Y_1(e^{j\omega})|'), xlabel('\omega/\pi'), axis([0,1,0,1])

w0=downsample(v,M); % Down-sampling without phase offset

W0=fft(w0,1024); % Computing the signal spectrum

subplot(3,2,4), plot(f,abs(W0(1:512))), text(0.8,0.7,'(e)')

ylabel('|W_0(e^{j\omega})|'), axis([0,1,0,1])

w1 = downsample(v,M,1); % Down-sampling with the phase offset

W1=fft(w1,1024); % Computing the signal spectrum

subplot(3,2,6), plot(f,abs(W1(1:512))), text(0.8,0.7,'(f)')

ylabel('|W_1(e^{j\omega})|'), xlabel(' \omega/\pi'), axis([0,1,0,1])

disp('END OF EXAMPLE 2.8')

END OF EXAMPLE 2.8

Comment: The plots demonstrate that the magnitude spectrum of the down sampled signal is independent of the phase offset when the frequency band of the original signal is properly bandlimited to , see Figs. (a), (b) and (c). On the contrary, the right-hand side shows different spectra of the downsampled signals for k = 0 (Fig. (e)) and k=1 (Fig. (f)). Evidently, the ftequency band of the original signal that exceeds , Fig. (d) results in different magnitude spectra of the down-sampled signals.

2. 8 Polyphase Decomposition and Reconstruction

• Polyphase components x0, x1, x2, x3 are obtained by down-sampling the original signal x with the phase offset.
• In the process of signal reconstruction, up-sampling with the phase offset over x0, x1, x2, x3 is used.

Example 2.9

y0 = upsample(x0,4);

y1 = upsample(x1,4,1);

y2 = upsample(x2,4,2);

y3 = upsample(x3,4,3);

figure (1)

subplot (4,1,1), stem(0:length(x0)-1,x0),ylabel('x_0[m]')

title('Polyphase komponents')

subplot (4,1,2), stem(0:length(x1)-1,x1),ylabel('x_1[m]')

subplot (4,1,3), stem(0:length(x2)-1,x2),ylabel('x_2[m]')

subplot (4,1,4), stem(0:length(x3)-1,x3),ylabel('x_3[m]')

xlabel('Time index m')

y = y0 + y1 + y2 + y3; % Adding together up-sampled polyphase components

figure (2)

subplot(2,1,1),stem(n,x), ylabel('x[n]'), axis([0,63,0,1])

title('Original signal')

subplot(2,1,2),stem(n,y), ylabel('y[n]'), axis([0,63,0,1])

title('Reconstructed signal')

xlabel('Time index n')

disp('END OF EXAMPLE 2.9')

END OF EXAMPLE 2.9

2.9 Multistage Systems

When decimation factor M can be factored into K integers , instead of using a single filter and factor-of-M down-sampler the overall decimator can be implemented as a cascade of K decimators as shown bellow.

The transfer function of the equivalent single-stage decimation filter can be obtained by applying the third identity to the above implementation scheme:

Example 2.10

clear all, close all

h1 = fir1(30,0.5); % FIR filter H1(z)

[H1,f] = freqz(h1,1,512,2);

figure

subplot(4,1,1), plot(f,abs(H1)), ylabel('|H_1(e^{j\omega})|')

axis([0,1,0,1.2])

M = 2;

h2 = zeros(1,M*length(h1));

h2([1:M:length(h2)]) = h1; % H2(z) = H1(z^2)

[H2,f] = freqz(h2,1,512,2);

subplot(4,1,2), plot(f,abs(H2)), ylabel('|H_1(e^{j2\omega})|')

axis([0,1,0,1.2])

M = 4;

h3 = zeros(1,M*length(h1));

h3([1:M:length(h3)]) = h1; % H3(z) = H1(z^4)

[H3,f] = freqz(h3,1,512,2);

subplot(4,1,3), plot(f,abs(H3)), ylabel('|H_1(e^{j4\omega})|')

axis([0,1,0,1.2])

H = H1.*H2.*H3; % Equivalent filter

subplot(4,1,4), plot(f,abs(H)),ylabel('|(H(e^{j\omega})|')

xlabel('\omega/\pi'), axis([0,1,0,1.2])

disp('END OF EXAMPLE 2.10')

END OF EXAMPLE 2.10

disp(' END OF CHAPTER II')

END OF CHAPTER II